When trying to solve inequalities (e.g. 1/(x+2)>x/(x-3)) I keep getting the wrong solutions even though my algebra is correct.

*The usual mistake with inequalities is not with a students algebra, but with lack of consideration of the inequality signs. During a tutorial I would show their false method and my correct one and also sketch (or get them to sketch) the graphs of both sides so they know how to check their answers.

When solving inequalities remember that whenever you multiply or divide both sides by a negative number you must flip the sign. So -2x > 3 means our solution is x < -3/2 NOT x > -3/2. So solving inequalities is not a simple as solving equations. So far simple, but what if we need to multiply both sides by something involving x? Taking your example 1/(x+2) > x/(x-3), we want to multiply both sides by (x+2)(x-3).

We do not know if this is positive or negative. Our trick here is to multiply by ((x+2)(x-3))²  which is ALWAYS positive, so the inequality sign does not change. We can now solve this algebraically. This technique can be used for any inequality, just multiply both sides by what you really want to multiply by SQUARED.